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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A one-level linear FAS correction is the Galerkin correction

Statement

Take $A = (\begin{array}{cc} 2 & - 1 \\ - 1 & 2 \end{array})$ , $P = (1, 1)^{T}$ , $I u = (u_{1} + u_{2}) / 2$ , $R = P^{T}$ , and the Galerkin coarse matrix $A_{H} = P^{T} A P = 2$ . For any fine iterate $u$ and right side $f$ , the coarse FAS difference $v - I u$ is exactly $A_{H}^{- 1} R (f - A u)$ , and the prolongated update is the ordinary Galerkin coarse correction.

Facts & Assumptions

Given: The two-dimensional linear system, the specified transfers, exact coarse solution, and exact real arithmetic.

[F1]

Finite-budget nonlinear FAS two-grid cycle: FAS solves the tau-corrected coarse equation and corrects by $u + P (v - I u)$ .

[F2]

Linear FAS exactly reduces to a Galerkin correction V/W cycle: for a linear Galerkin hierarchy with $R = P^{T}$ , the full recursive FAS cycle is the same as the linear correction V/W cycle.

Example

Given: The matrix $A$ , prolongation $P$ , restriction $I$ , and residual restriction $R$ in the Statement.

Verification

1.1givenF1algebra

Direct multiplication gives $A P = (1, 1)^{T}$ and $A_{H} = P^{T} A P = 2$ . Write $m = I u$ and $r = f - A u$ . The FAS coarse target is $A_{H} m + R r = 2 m + r_{1} + r_{2} .$ The exact coarse solution is therefore $v = m + (r_{1} + r_{2}) / 2$ .

2.1F1F2step 1.1algebra∎

Since $A_{H}^{- 1} R r = (r_{1} + r_{2}) / 2$ , the coarse difference is $v - I u = A_{H}^{- 1} R (f - A u)$ and the fine update is $u + P (v - I u) = u + P A_{H}^{- 1} P^{T} (f - A u),$ the standard Galerkin correction. With matching residual-correction smoothers, [F2] identifies the full V/W iterate as well.

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